On the role of tail in stability and energetic cost of bird flapping flight

Migratory birds travel over impressively long distances. Consequently, they have to adopt flight regimes being both efficient—in order to spare their metabolic resources—and robust to perturbations. This paper investigates the relationship between both aspects, i.e., energetic performance and stability, in flapping flight of migratory birds. Relying on a poly-articulated wing morphing model and a tail-like surface, several families of steady flight regime have been identified and analysed. These families differ by their wing kinematics and tail opening. A systematic parametric search analysis has been carried out, in order to evaluate power consumption and cost of transport. A framework tailored for assessing limit cycles, namely Floquet theory, is used to numerically study flight stability. Our results show that under certain conditions, an inherent passive stability of steady and level flight can be achieved. In particular, we find that progressively opening the tail leads to passively stable flight regimes. Within these passively stable regimes, the tail can produce either upward or downward lift. However, these configurations entail an increase of cost of transport at high velocities penalizing fast forward flight regimes. Our model-based predictions suggest that long range flights require a furled tail configuration, as confirmed by field observations, and consequently need to rely on alternative mechanisms to stabilize the flight.

IBIS S1 Table S1 -Morphological and kinematic parameters of the northen bald ibis The bird parameters to capture the morphology of the northern bald ibis are reported in Table S1. The data related to the bird morphology are listed in Table S1(a). The parameters governing the wing kinematics after solving Equation 2 are reported in Table S1(b).    [1,2,3,4]. The choice of the length of these segments relies on the linear regression in log − log plot comparing the available data as a function of the mass of the species collected.
As stated in the manuscript, the kinematics is governed by sinusoidal trajectories in the form Sinusoidal trajectories are a simplistic approximation of actual kinematics. However, despite possible inaccuracies, this choice allows us to drastically reduce the number of parameters involved in the problem, but yet permits to reproduce flapping gaits exploiting all its features (sweep, folding, etc.).

S1 TABLE S1 -MORPHOLOGICAL AND KINEMATIC PARAMETERS OF THE NORTHEN BALD IBIS
The kinematic parameters of Equation 1 were tuned as follows. We assumed the bird having its largest wing extension in the downstroke phase, and its minimum extension during the upstroke [5,6]. This wing extension/folding is regulated via the kinematics equation about the z −axis, and thus imposing the relative phase of the three joints as Tobalske [5]. The values of A s,z is kept constant to a value of 14 • , which suffice enough nose-up and nose-down moment to assure limit cycle conditions in absence of the tail surface.
As last, the relative phase of the joints governing the wing pitch, and thus the rotation of the wing profiles about the y−axis is modeled in such a way the wing is pitching up during the upstroke, and pitching down during the downstroke, in order to reduce the variations of the angle of attack during the whole flapping period.
From a mathematical point of view, this implies setting a phase of where i represents the shoulder and wrist joints.
The resulting kinematics -and the wingtip trajectory -are reported for the three views of the bird in Figure S1, S2, S3. S1

S2 Additional files
In order to reproduce all the Figures presented in the paper, we provide the available data of the simulations, as well as the Python scripts. The file are are zipped in the supplementary files folder, and the listing is here reported: • dataset.csv: dataset of the detected limit cycles in .csv form. It is structured as follows: where: β: tail opening; ψ s,z : sweep offset of the shoulder -A s,x : wingbeat amplitude; -CoT: cost of transport; -U f f : resulting forward flight velocity in the inertial frame; -W f f : resulting vertical velocity in the inertial frame; -P: power; -Λ max : largest Floquet multiplier; q w,y : resulting elbow offset; θ: resulting body pitch angle; -M w : moment of the wing; -M t : moment of the tail.
• tail x sweep y SAz z: Folders. They contain the limit cycle solutions, the gust simulations, and the aerodynamic moments to reproduce the three cases reported in Section Comparison between furled and open tail solutions. The nomenclature is named accordingly to each study case.
• figure 4.py: Python script. It reads the three limit cycles solutions, and produces Figure 4.
• figure 5.py: Python script. It reads the gust simulations, and produces Figure 5.
All the attached files are standalone and commented accordingly to help the readability. We remind the user to check the proper path-to-file in order to read the different dataset on the local machine.